Remark 1 (Ensemble transform particle filter) 
In his pioneering approach to filtering [1]}, Reich proposes a nonparametric estimator for Monge's map, by solving the Kantorovich relaxation of a discrete optimal transport problem
between the empirical measure given by the prior ensemble and
the empirical approximation of the
posterior distribution
obtained by reweighting prior
particles according to their likelihood. In this paper, we
consider continuous rather than discrete transport, and look for
prior-to-posterior transformations—not necessarily optimal with respect to any cost—that can
be estimated without resorting to importance sampling, thus avoiding
the root cause of particle
degeneracy.

